# sphinx-proof: advanced proofs, theorems & more...

Easily render proofs, theorems, axioms, lemmas, definitions and much more with `sphinx-proof`.
Read more in the [documentation of sphinx-proof](sphinxproof:syntax).

````{prf:proof}
We'll omit the full proof.

But we will prove sufficiency of the asserted conditions.

To this end, let $y \in \mathbb R^n$ and let $S$ be a linear subspace of $\mathbb R^n$.

Let $\hat y$ be a vector in $\mathbb R^n$ such that $\hat y \in S$ and $y - \hat y \perp S$.

Let $z$ be any other point in $S$ and use the fact that $S$ is a linear subspace to deduce

```{math}
\| y - z \|^2
= \| (y - \hat y) + (\hat y - z) \|^2
= \| y - \hat y \|^2  + \| \hat y - z  \|^2
```

Hence $\| y - z \| \geq \| y - \hat y \|$, which completes the proof.
````

[sphinxproof:syntax]: https://sphinx-proof.readthedocs.io/en/latest/index.html "documentation of sphinx-proof"